Diophantine Equations

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Diophantine Equations are a set of one or more equations in which only whole number solutions are allowed. This subfield of mathematics is named after a famous ancient Greek mathematician Diophantus. Diophantine problems involve solving a set of equations for more than one variable, but the solutions must be whole numbers. Here is an example of a Diophantine problem:

This problem can be written as the following two simultaneous equations in three variables. Just remember that the solution must be whole numbers.

Compared with other Diophantine problems, this is a rather easy one. It has only one solution. (Have you figured it out yet?) If you don't know of any easy way to solve it, then you can always resort to trying all possible values for c, d, and t between 0 and 60, and seeing which values satisfy both equations. Better yet, you might write a short computer program to do it for you.

There are many other Diophantine equations that are much more difficult that this one. A classic example is the following:

As long as n = 1 or n = 2, then you can find infinitely many values for x, y, and z that solve the equation. (For example, 2¹ + 3¹ = 5¹, or 3² + 4² = 5²)
But, are there any solutions if n > 2?

This is a famous problem known as Fermat's Last Theorem named after Pierre de Fermat. In 1637, Fermat wrote in his copy of the book Arithmetica, (a book written by Diophantus), that this equation has no solution in positive integers if n > 2, and that he just thought of a brilliant way to prove it, but that the margin of the book is too small for him to write it down.

After Fermat's death, many people, including many of the world's most famous mathematicians, tried to prove Fermat's Last Theorem, but were not able to do so for over 300 years. In fact, some were beginning to speculate that perhaps this was an example of a theorem that could never be proved or disproved, as predicted by Gödel's Incompleteness Theorem, until Andrew Wiles actually published a proof of it in 1995.

By the way, in case you're still wondering about the answer to the problem where you were asked how many birds the poultry farmer sold, there is only one answer: c = 51, d = 8, t = 1.

However, if the poultry farmer had told you that instead of $292 he sold them for $319, then there are two solutions: c = 39, d = 20, t = 1; and c = 46, d = 4, t = 10.

On the other hand, if the farmer said that he sold the 60 birds for $293, then you can call him a liar because there’s no way that he could have sold those 60 birds for $293.

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